Hypergraphs
and their planar embeddings
Marisa Debowsky
University of Vermont
April 25, 2003
Things I Want You To
Get Out Of This Lecture




The definition of a hypergraph.
Some understanding of the main
question: “When is a hypergraph
planar?”
The concept of a partial ordering
on graphs.
Some understanding of the answer
to the main question!
Definitions


A hypergraph is a generalization of a graph.
An edge in a graph is defined as an
(unordered) pair of vertices. In a hypergraph,
an edge (or hyperedge) is simply a subset of
the vertices (of any size).
The rank of a hyperedge is the number of
vertices incident with that edge. The rank of
the hypergraph H is the size of the largest
edge of H.
Example
1
2
4
3
5
6
V(H) = {1, 2, 3, 4, 5, 6}
E(H) = {124, 136, 235, 456}
Planar Graphs


A graph G is planar if there exists a drawing of
G in the plane with no edge crossings.
Kuratowski gave necessary and sufficient
conditions for a graph to be planar:
Thm: A graph G is planar if, and only if, it
contains no subdivision of K3,3 or K5.
Planar Hypergraphs?

In order to ask questions about
planar hypergraphs, we need to
make sure that the concept is
well-defined.
Drawing a Hypergraph
the long-winded definition
Defn: A hypergraph H has an embedding (or is planar) if
there exists a graph M such that V(M) = V(H) and M can
be drawn in the plane with the faces two-colored (say,
grey and white) so that there exists a bijection between
the grey faces of M and the hyperedges of H so that a
vertex v is incident with a grey face of M iff it is incident
with the corresponding hyperedge of H.
Example
V(M) = {1, 2, 3, 4, 5, 6}
E(M) = {12, 24, 14, 13, 36, 16,
23, 25, 35, 45, 56, 46}
F(M) = {124, 136, 235, 456,
123, 245, 356, 146}
1
2
4
3
5
6
V(H) = {1, 2, 3, 4, 5, 6}
E(H) = {124, 136, 235, 456}
F(H) = {123, 245, 356, 146}
Main Question

Which hypergraphs are planar?
Can we find an obstruction set to
planar hypergraphs (akin to K3,3
and K5 for planar graphs)?
(Okay, that was more than one question.)
The Incidence Graph
Given a hypergraph, H, we can construct a bipartite
graph G derived from H.
Let V1È V2 be the vertices of G. The vertices in V1
correspond to V(H) and the vertices in V2 correspond
to E(H). A vertex v Î V1 is adjacent to a vertex w Î V2
if the corresponding hypervertex v is incident with the
corresponding hyperedge w. Because the bipartite
graph describes the incidences of the vertices and
edges of H, we call G the incidence graph of H.
Example
1
2
4
1
3
5
2
6
4
3
5
6
In the bipartite graph on the right, the
circled vertices correspond to hyperedges.
A Handy Reduction Theorem
and the Main Question, again
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Thm: A hypergraph is planar if and
only if its incidence graph is planar.
This allows us to rephrase our
question:
Which bipartite graphs are planar?
Graphs Inside Graphs
When we say that K3,3 and K5 are the “smallest”
non-planar graphs or the “obstructions” to planarity,
we mean that every non-planar graph contains a
copy of K3,3 or K5 as a subgraph - in other words,
contains of subdivision of K3,3 or K5.
Can we formulate a notion similar to “subgraph” or
“subdivision” for bipartite graphs that extends
naturally to hypergraphs?
Partial Orderings

We can rank graphs using a
partially ordered set: the set of all
graphs together with a relation “< ”
which is reflexive, antisymmetric,
and transitive.
Note: This is different from
a “totally ordered set”!
Graph Operations
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Frequently, we will form a graph G2 from a graph
G1 where G2 < G1 by a modification called a
graph operation. Different combinations of
operations create distinct partial orderings of
graphs. You are already familiar with some:
deleting an edge from G1, for example, creates a
subgraph of G1.
We will consider four different partial orders:
detachment, bisubdivision, deflation, and duality.
Hereditary Properties


A property P is called hereditary
under the partial order “ < ” if,
whenever GÎ P and H < G, it
follows that H Î P.
Planarity is a hereditary property
under these four operations, so we
can consider the obstruction set to
planarity under each operation.
Size of the Obstruction Sets
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The detachment operation on
hypergraphs corresponds to the
subgraph operation in graphs: its
obstruction set is infinite.
Adding the bisubdivision operation
reduces the obstructions to a finite
set, and each additional operation
makes the set smaller.
Detachment Ordering

H is a detachment of G if it is
obtained by removing an edge
from the incidence graph. This
corresponds to removing an
incidence between a vertex and a
hyperedge: pictorally, “detaching” a
vertex from the hyperedge.
Under the detachment ordering,
H < G iff H is a detachment of G.
Detachment Example
Bisubdivision Ordering

H is a bisubdivision of G if it is
formed by removing two interior
degree-2 vertices from an edge
of the incidence graph. This
corresponds to contracting a
hyperedge of rank 2.
Under the bisubdivision ordering,
H < G iff H is a bisubdivision or
detachment of G.
Bisubdivision Example
Deflation Ordering

Suppose a bipartite graph G has a vertex of degree
n from one partite set surrounded by (that is,
adjacent to) n vertices of degree 2 from the other
partite set. H is a deflation of G if it is obtained by
removing those n vertices and reassigning the
interior vertex (still of degree n) to the other partite
set. In the hypergraph, this corresponds to
“deflating” a hyperedge of rank n to a single vertex.
Under the deflation ordering, H < G iff H is a
deflation, bisubdivision, or detachment of G.
Deflation Example
Duality
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
The incidence graph is a bipartite graph; one
partite set corresponds to the vertices of the
hypergraph and the other to the hyperedges.
Reversing the assignments of the partite sets
produces a (generally) different hypergraph.
Defn: A hypergraph H is the dual of a
hypergraph G if they are obtained from the
same incidence graph.
Duality Ordering
and Example

The duality ordering has H < G iff
H is the dual of G.
Bipartite Incidence Graph
Hypergraph G
Hypergraph H
The Main Question... Again.

One more time:
What are the obstructions to
embedding bipartite graphs in the
plane under each partial ordering?
The Answer!
(for bipartite graphs)

Thm: There are exactly 9 nonplanar bipartite graphs under the
partial ordering of bisubdivision
and detachment.
The bipartite obstructions,
G1 - G9, are given below.
Bipartite graphs G1 - G9
G
G
G
1
2
3
G4
G5
G6
G7
G8
G
9
The Answer!
(for hypergraphs)

Corollary: There are exacly 16
non-planar hypergraphs under the
partial ordering of bisubdivision
and detachment.
The hypergraph obstructions,
H1 - H16, are given below.
H1
H2
H
H4
3
H
H
H
H
5
6
7
8
H11
H1
H
H1
9
0
H1
3
2
H14
H15
H16
Other Partial Orderings
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Thm: There are exactly 2 non-planar
bipartite graphs under the partial ordering of
deflation, bisubdivision, and detachment.
They are G1 and G4.
Corollary: There are exactly 3 non-planar
hypergraphs under the partial ordering of
deflation, bisubdivision, and detachment.
They are H1, H2, and H7.
Still More Partial Orderings

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Thm: There are exactly 9 non-planar
hypergraphs under the partial ordering of
duality, bisubdivision, and detachment.
They are H1, H3, H5, H7, H8, H9, H11, H13
and H15.
Thm: There are exactly 2 non-planar
hypergraphs under the partial ordering of
duality, deflation, bisubdivision, and
detachment. They are H1 and H7.
Further Research
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Analogues of Kuratowski’s Theorem have been
developed for other surfaces. Can we find the
obstruction sets for embedding hypergraphs in,
for example, the projective plane?
There are 2 non-planar graphs and 16 nonplanar hypergraphs. There are 103 nonprojective-planar graph, which leads us to
suspect on the order of 800 non-projectiveplanar hypergraphs.
Contact Information

You can reach me at
[email protected]
or find me online at
http://www.emba.uvm.edu/~mdebowsk/.

The work presented was done jointly with
Professor Dan Archdeacon at UVM. You can
reach him at [email protected].
H1
H5
H9
H13
H2
H3
H6
H10
H14
H4
H7
H8
H11
H12
H15
H16